Three concrete equivalences

Task number: 3394

Determine whether the following relations are equivalences and if they are, determine their equivalence classes.

  • Variant

    \(X_1={\mathbb N}, xR_1y \Leftrightarrow p |(x-y)\) (residue classes modulo \(p\in {\mathbb N}, p\ge 2\))

  • Variant

    \(X_2={\mathbb Z}\setminus 0, xR_2y \Leftrightarrow x|y \wedge y|x\)

  • Variant

    \(X_3={\mathbb N}, xR_3y \Leftrightarrow \exists z\in {\mathbb N}: z|y \wedge z|x\).

    What happens if we require \(z>1\)?

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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